General Game Playing Michael Thielscher, Dresden Some of the - PowerPoint PPT Presentation

AAAI'08 Tutorial Chess Players General Game Playing Michael Thielscher, Dresden Some of the material presented in this tutorial originates in work by Michael Genesereth and the Stanford Logic Group. We greatly appreciate their contribution. The Turk (18 th Century) Alan Turing & Claude Shannon (~1950)

Deep-Blue Beats World Champion (1997) In the early days, game playing machines were considered a key to Artificial Intelligence (AI). But chess computers are highly specialized systems. Deep-Blue's intelligence was limited. It couldn't even play a decent game of Tic-Tac-Toe or Rock-Paper-Scissors. With General Game Playing many of the original expectations with game playing machines get revived. Traditional research on game playing focuses on constructing specific evaluation functions A General Game Player is a system that building libraries for specific games understands formal descriptions of arbitrary strategy games The intelligence lies with the programmer, not with the program! learns to play these games well without human intervention A General Game Player needs to exhibit much broader intelligence: abstract thinking strategic planning learning

� General Game Playing and AI Rather than being concerned with a specialized solution to a narrow problem, General Game Playing encompasses a variety of AI areas: Game Playing Games Agents Knowledge Representation Deterministic, complete information Competitive environments Nondeterministic, partially observable Uncertain environments Planning and Search Rules partially unknown Unknown environment model Learning Robotic player Real-world environments General Game Playing is considered a grand AI Challenge Application (1) Application (2): Economics Commercially available chess computers can't be used for a game of Bughouse Chess. A General Game Playing system could be used for negotiations, marketing strategies, pricing, etc. It can be easily adapted to changes in the business processes and rules, new competitors, etc. The rules of an -marketplace can be formalized as a game, so that agents can automatically learn how to participate. An adaptable game computer would allow the user to modify the rules for arbitrary variants of a game.

Single-Player, Deterministic Example Games Single-Player, Deterministic Two-Player, Zero-Sum, Deterministic

Two-Player, Zero-Sum, Deterministic Two-Player, Zero-Sum, Nondeterministic n -Player, Deterministic n -Player, Incomplete Information, Nondeterministic

General Game Playing Initiative Roadmap The Game Description Language GDL: games.stanford.edu Knowledge Representation Game description language How to make legal moves: Variety of games/actual matches Automated Reasoning Basic player available for download How to solve simple games: Annual world cup @AAAI (since 2005) Planning & Search Price money: US$ 10,000 How to play well: Learning (deterministic games w/ complete information only) Every finite game can be modeled as a state transition system Game Description Language But direct encoding impossible in practice 19,683 states ~ 10 43 legal positions

✁ ✁ Modular State Representation: Fluents Fluent Representation for Chess (2) 8 8 7 cell(X,Y,C) 7 canCastle(P,S) 6 6 P ∈ { white,black } X ∈ { a, ... ,h } 5 5 S ∈ { kingsSide,queensSide } Y ∈ { 1, ... ,8 } 4 4 C ∈ { whiteKing, ... ,blank } enPassant(C) 3 3 control(P) C ∈ { a, ... ,h } 2 2 P ∈ { white,black } 1 1 a b c d e f g h a b c d e f g h Actions Game Rules (I) 8 Players roles([white,black]) 7 move(U,V,X,Y) U,X ∈ { a, ... ,h } 6 V,Y ∈ { 1, ... ,8 } 5 4 Initial position init(cell(a,1,whiteRook)) ... promote(X,Y,P) 3 X,Y ∈ { a, ... ,h } 2 P ∈ { whiteQueen, ...} 1 legal(white,promote(X,Y,P)) <= a b c d e f g h Legal Moves ... true(cell(X,7,whitePawn))

✂ ✁ ✂ ✁ ✂ ✁ ✁ Game Rules (II) Clausal Logic Variables: X, Y, Z Constants: a, b, c next(cell(X,Y,C)) <= Position updates does(P,move(U,V,X,Y)) Functions: f, g, h true(cell(U,V,C)) Predicates: p, q, r, = Logical Operators: ¬, , , <= terminal <= End of game Terms: X, Y, Z, a, b, c, f(a), g(a,X), h(a,b,f(Y)) checkmate stalemate Atoms: p(a,b) Literals: p(a,b), ¬q(X,f(a)) goal(white,100) <= true(control(black)) Clauses: Head <= Body Result checkmate Head : relational sentence goal(white, 50) <= stalemate Body : logical sentence built from , , literal Game-Independent Vocabulary Axiomatizing Tic-Tac-Toe: Fluents Relations cell(X,Y,M) 3 roles(list-of(player)) X,Y ∈ { 1,2,3 } init(fluent) M ∈ { x,o,b } true(fluent) 2 does(player,move) next(fluent) control(P) 1 legal(player,move) P ∈ { xplayer,oplayer } goal(player,value) 1 2 3 terminal

✁ ✁ ✁ Axiomatizing Tic-Tac-Toe: Actions Tic-Tac-Toe: Vocabulary Constants Players xplayer, oplayer Marks x, o, b 3 Functions mark(X,Y) cell(number,number,mark) Fluent Fluent control(player) 2 X,Y ∈ { 1,2,3 } Action mark(number,number) Predicates noop 1 row(number,mark) column(number,mark) 1 2 3 diagonal(mark) line(mark) open Players and Initial Position Preconditions roles([xplayer,oplayer]) legal(P,mark(X,Y)) <= init(cell(1,1,b)) true(cell(X,Y,b)) true(control(P)) init(cell(1,2,b)) init(cell(1,3,b)) legal(xplayer,noop) <= init(cell(2,1,b)) true (cell(X,Y,b)) true(control(oplayer)) init(cell(2,2,b)) init(cell(2,3,b)) legal(oplayer,noop) <= init(cell(3,1,b)) true (cell(X,Y,b)) init(cell(3,2,b)) true(control(xplayer)) init(cell(3,3,b)) init(control(xplayer))

✁ ✁ ✁ ✁ ✁ ✂ ✁ ✁ ✁ ✂ ✁ ✁ ✁ ✁ ✁ ✁ ✁ Update Termination next(cell(M,N,x))<= does(xplayer,mark(M,N)) terminal <= line(x) line(o) next(cell(M,N,o))<= does(oplayer,mark(M,N)) terminal <= ¬open next(cell(M,N,W))<= true(cell(M,N,W)) ¬W=b line(W) <= row(M,W) next(cell(M,N,b))<= true(cell(M,N,b)) line(W) <= column(N,W) does(P,mark(J,K)) (¬M=J ¬N=K) line(W) <= diagonal(W) next(control(xplayer)) <= true(control(oplayer)) open <= true(cell(M,N,b)) next(control(oplayer)) <= true(control(xplayer)) Supporting Concepts Goals row(M,W) <= true (cell(M,1,W)) true (cell(M,2,W)) true(cell(M,3,W)) goal(xplayer,100) <= line(x) goal(xplayer,50) <= ¬ line(x) ¬ line(o) ¬open column(N,W) <= true (cell(1,N,W)) goal(xplayer,0) <= line(o) true (cell(2,N,W)) true(cell(3,N,W)) goal(oplayer,100) <= line(o) goal(oplayer,50) <= ¬ line(x) ¬ line(o) ¬open diagonal(W) <= true (cell(1,1,W)) true (cell(2,2,W)) goal(oplayer,0) <= line(x) true(cell(3,3,W)) diagonal(W) <= true (cell(1,3,W)) true (cell(2,2,W)) true(cell(3,1,W))

Finite Games Games as State Machines Finite Environment b e h Game “world” with finitely many states One initial state and one or more terminal states Fixed finite number of players Each with finitely many “percepts” and “actions” Each with one or more goal states a c f i k Causal Model Environment changes only in response to moves Synchronous actions d g j Initial State and Terminal States Simultaneous Actions a/a a/b b e h b e h b/b a/a a/b a/a a/a b/b b/a a/b a/b a c f i k a/b a c f i k a/a a/b b/a b/a b/b b/b a/a a/a d g j d g j

✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✝ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✄ ✞ ✞ ☎ ✞ ✄ ✆ ✞ ✞ ✄ ✞ ✞ ✞ GDL for Trading Games: Example Game Model (English Auction) An n -player game is a structure with components: role(bidder_1) ... role(bidder_n) next(winner(P)) <= does(P,bid(X)) bestbid(X) S – set of states next(highestBid(X)) <= does(P,bid(X)) bestbid(X) init(highestBid(0)) next(winner(P)) <= true(winner(P)) not bid A 1 , ..., A n – n sets of actions, one for each player init(round(0)) next(highestBid(X)) <= true(highestBid(X) not bid next(round(N)) <= true(round(M)), successor(M,N) legal(P,bid(X)) <= l 1 , ..., l n – where l i A i × S, the legality relations true(highestBid(Y)) greaterthan(X,Y) bid <= does(P,bid(X)) legal(P,noop) u: S × A 1 × ... × A n S – update function bestbid(X) <= does(P,bid(X)) not overbid(X) terminal <= true(round(10)) overbid(X) <= does(P,bid(Y)) greaterthan(Y,X) S – initial game state s 1 t S – the terminal states g 1 , ... g n – where g i S × , the goal relations Try it Yourself: Play this Game! role(you) legal(you,jump(X,Y)) <= true(cell(X,onecoin)) true(cell(Y,onecoin)) init(step(1)) ( twobetween(X,Y) | twobetween(Y,X) ) init(cell(1,onecoin)) init(cell(Y,onecoin)) <= succ(X,Y) zerobetween(X,Y) <= succ(X,Y) succ(1,2) succ(2,3) ... succ(7,8) zerobetween(X,Y) <= succ(X,Z) true(cell(Z,zerocoins)) Automated Reasoning zerobetween(Z,Y) next(step(Y)) <= true(step(X)) succ(X,Y) onebetween(X,Y) <= succ(X,Z) true(cell(Z,zerocoins)) next(cell(X,zerocoins)) <= does(you,jump(X,Y)) onebetween(Z,Y) next(cell(Y,twocoins)) <= does(you(jump(X,Y)) onebetween(X,Y) <= succ(X,Z) true(cell(Z,onecoin)) next(cell(X,C)) <= does(you,jump(Y,Z)) zerobetween(Z,Y) true(cell(X,C)) twobetween(X,Y) <= succ(X,Z) true(cell(Z,zerocoins)) distinct(X,Y) distinct(X,Z) twobetween(Z,Y) twobetween(X,Y) <= succ(X,Z) true(cell(Z,onecoin)) terminal <= ~continuable onebetween(Z,Y) continuable <= legal(you,M) twobetween(X,Y) <= succ(X,Z) true(cell(Z,twocoins)) goal(you,100) <= true(step(5)) zerobetween(Z,Y) goal(you,0) <= true(cell(X,onecoin))